nLab
essential geometric morphism

Idea

A geometric morphism $f:E\to F$ between toposes is a functor of the underlying categories that is consistent with the interpretation of $E$ and $F$ as generalized topological spaces.

If $F=\mathrm{Set}=\mathrm{Sh}(*)$ is the terminal sheaf topos, then $E\to \mathrm{Set}$ is essential if $E$ is a locally connected topos . In general, $f$ being essential is a necessary (but not sufficient) condition to ensure that $f$ behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.

Definition

Definition

Given a geometric morphism $(f^{*}⊣f_{*}):E\to F$, it is an essential geometric morphism if the inverse image functor $f^{*}$ has not only the right adjoint $f_{*}$, but also a left adjoint $f_{!}$:

$$(f_{!}⊣f^{*}⊣f_{*}):E\stackrel{\stackrel{f_{!}}{\to }}{\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}}F.$$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{f_!}{\to}}{\stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}} F \,.

A point of a topos $x:\mathrm{Set}\to E$ which is given by an essential geometric morphism is called an essential point of $E$.

Remark

There are various further conditions that can be imposed on a geometric morphism:

• If $f_{!}$ can be made into an $E$-indexed functor and $f^{*}$ satisfies some extra conditions, the geometric morphism $f$ is a locally connected geometric morphism (see there for details).

• If $f_{!}$ preserves finite products then $f$ is called connected surjective.

• If $f_{!}$ preserves finite products and moreover there is a further functor $f^{!}:F\to E$ which is right adjoint $(f_{*}⊣f^{!})$ and full and faithful, i.e. if we have a sequence of adjunctions

  $$(f_{!}⊣f^{*}⊣f_{*}⊣f^{!})$$(f_! \dashv f^* \dashv f_* \dashv f^!)

  with full and faithful $f^{!}$ then the geometric morphism $f$ is called local geometric morphism.

  In this case in particular $F\stackrel{\stackrel{p_{*}}{\leftarrow }}{\underset{p^{!}}{\hookrightarrow }}E$

  is a geometric embedding and hence makes $F$ a subtopos of $E$.

Properties

Relation to morphisms of (co)sites

For $C$ and $D$ small categories write $[C,\mathrm{Set}]$ and $[D,\mathrm{Set}]$ for the corresponding copresheaf toposes. (If we think of the opposite categories $C^{\mathrm{op}}$ and $D^{\mathrm{op}}$ as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)

Proposition

This construction extends to a 2-functor

$$[-,\mathrm{Set}]:{\mathrm{Cat}}_{\mathrm{small}}^{\mathrm{co}}\to {\mathrm{Topos}}_{\mathrm{ess}}$$[-,Set] : Cat_{small}^{co} \to Topos_{ess}

from the 2-category Cat${}_{\mathrm{small}}$ with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor $f:C\to D$ is sent to the essential geometric morphism

$$(f_{!}⊣f^{*}⊣f_{!}):[C,\mathrm{Set}]\stackrel{\stackrel{f_{!}:={\mathrm{Lan}}_f}{\to }}{\stackrel{\stackrel{f^{*}:=(-)\circ f}{\leftarrow }}{\underset{f^{*}:={\mathrm{Ran}}_f}{\to }}}[D,\mathrm{Set}],$$(f_! \dashv f^* \dashv f_!) : [C,Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f^* := Ran_f}{\to}}} [D,Set] \,,

where ${\mathrm{Lan}}_f$ and ${\mathrm{Ran}}_f$ denote the left and right Kan extension along $f$, respectively.

Proposition

This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:

$$[-,\mathrm{Set}]:{\mathrm{Cat}}_{\mathrm{CauchyComp}}^{\mathrm{co}}\hookrightarrow {\mathrm{Topos}}_{\mathrm{ess}}.$$[-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess} \,.

For all small categories $C,D$ we have an equivalence of categories

$$\mathrm{Func}(\overline{C},\overline{D}{)}^{\mathrm{op}}\simeq {\mathrm{Topos}}_{\mathrm{ess}}([C,\mathrm{Set}],[D,\mathrm{Set}])$$Func(\overline{C},\overline{D})^{op} \simeq Topos_{ess}([C,Set], [D,Set])

between the opposite category of the functor category between the Cauchy completions of $C$ and $D$ and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.

In particular, since every poset – when regarded as a category – is Cauchy complete, we have

Corollary

The 2-functor

$$[-,\mathrm{Set}]:\mathrm{Poset}\to {\mathrm{Topos}}_{\mathrm{ess}}$$[-,Set] : Poset \to Topos_{ess}

is a full and faithful 2-functor.

Remark

Sometimes it is useful to decompose this statement as follows.

There is a functor

$$\mathrm{Alex}:\mathrm{Poset}\to \mathrm{Locale}$$Alex : Poset \to Locale

which assigns to each poset a locale called its Alexandroff locale. By a theorem discussed there, a morphisms of locales $f:X\to Y$ is in the image of this functor precisely if its inverse image morphism $f^{*}\mathrm{Op}(Y)\to \mathrm{Op}(X)$ of frames has a left adjoint in the 2-category Locale.

Moreover, for any poset $P$ the sheaf topos over $\mathrm{Alex}P$ is naturally equivalent to $[P,\mathrm{Set}]$

$$[-,\mathrm{Set}]\simeq \mathrm{Sh}\circ \mathrm{Alex}.$$[-,Set] \simeq Sh \circ Alex \,.

With this, the fact that $[-,\mathrm{Set}]:\mathrm{Poset}\to \mathrm{Topos}$ hits precisely the essential geometric morphisms follows with the basic fact about localic reflection, which says that $\mathrm{Sh}:\mathrm{Locale}\to \mathrm{Topos}$ is a full and faithful 2-functor.

Some morphism calculus

Proposition

Let $f:E\to F$ be an essential geometric morphism.

For every $\varphi :X\to f^{*}{f}_{*}A$ in $E$ the diagram

$$\begin{array}{ccc}X& \stackrel{\varphi }{\to }& f^{*}{f}_{*}A\\ \downarrow & & \downarrow \\ f^{*}{f}_{!}X& \stackrel{}{\to }& A\end{array}$$\array{ X &\stackrel{\phi}{\to}& f^* f_* A \\ \downarrow && \downarrow \\ f^* f_! X &\stackrel{}{\to}& A }

commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of $\varphi$ under the composite adjunction $(f^{*}{f}_{!}⊣f^{*}{f}_{*})$.

Proof

The morphism $\varphi :X\to f^{*}{f}_{*}A$ is the component of a natural transformation

$$\begin{array}{ccccc}*& & \stackrel{X}{\to }& & E\\ {}^A\downarrow & {\Downarrow }^{\varphi }& & {}^{f^{*}}\nearrow \\ E& \underset{f_{*}}{\to }& F\end{array}.$$\array{ *&&\overset{X}{\to}&& E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow \\ E &\underset{f_*}{\to}&F } \,.

The composite $X\stackrel{\varphi }{\to }{f}^{*}{f}_{*}A\to A$ is the component of this composed with the counit $f^{*}{f}_{*}\Rightarrow \mathrm{Id}$.

We may insert the 2-identity given by the zig-zag law

$$\cdots =\begin{array}{ccccccccc}*& & \stackrel{X}{\to }& & E& & =& & E\\ {}^A\downarrow & {\Downarrow }^{\varphi }& & {}^{f^{*}}\nearrow & \Downarrow & {\searrow }^{f_{!}}& \Downarrow & {\nearrow }_{f^{*}}\\ E& \underset{f_{*}}{\to }& F& & =& & F\end{array}.$$\cdots \;\;\; = \;\;\; \array{ *&&\overset{X}{\to}&& E && = && E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!} &\Downarrow& \nearrow_{\mathrlap{f^*}} \\ E &\underset{f_*}{\to}&F &&=&& F } \,.

Composing this with the counit $f^{*}{f}_{*}\Rightarrow \mathrm{Id}$ produces the transformation whose component is manifestly the morphism $X\to f^{*}{f}_{!}X\to A$.

Examples

Etale geometric morphisms

For any morphism $f:A\to B$ in a topos $E$, the induced geometric morphism $f:E/A\to E/B$ of overcategory toposes is essential.

For the case $B=*$ the terminal object, the geometric morphism

$$\pi :E/A\to E$$\pi : E/A \to E

is also called an etale geometric morphism.

Locally connected toposes

A locally connected topos $E$ is one where the global section geometric morphism $\Gamma :E\to \mathrm{Set}$ is essential.

$$(f_{!}⊣f^{*}⊣f_{*}):E\stackrel{\stackrel{{\Pi }_0}{\to }}{\stackrel{\stackrel{\mathrm{LConst}}{\leftarrow }}{\underset{\Gamma }{\to }}}\mathrm{Set}.$$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{\Pi_0}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} Set \,.

In this case, the functor ${\Gamma }_{!}={\Pi }_0:E\to \mathrm{Set}$ sends each object to its set of connected components. More on this situation is at homotopy groups in an (∞,1)-topos.

Note, though that if $p:E\to S$ is an arbitrary geometric morphism through which we regard $E$ as an $S$-topos, i.e. a topos “in the world of $S$,” the condition for $E$ to be locally connected as an $S$-topos is not just that $p$ is essential, but that the left adjoint $p_{!}$ can be made into an $S$-indexed functor (which is automatically true for $p^{*}$ and $p_{*}$). This is automatically the case for $\mathrm{Set}$-toposes (at least, when our foundation is material set theory—and if our foundation is structural set theory, then our large categories and functors all need to be assumed to be $\mathrm{Set}$-indexing anyway). For more see locally connected geometric morphism.

Tiny objects

The tiny objects of a presheaf topos $[C,\mathrm{Set}]$ are precisely the essential points $\mathrm{Set}\to [C,\mathrm{Set}]$. See tiny object for details.

Related concepts

• essential geometric morphism

• locally connected topos / locally ∞-connected (∞,1)-topos

• connected topos / ∞-connected (∞,1)-topos

• totally connected topos / totally connected (∞,1)-topos

• local topos / local (∞,1)-topos.

• cohesive topos / cohesive (∞,1)-topos

References

The definition of geometric morphism appears before Lemma A.4.1.5 in

• Peter Johnstone, Sketches of an Elephant

Connected surjective and local geometric morphisms are discussed in

• Peter Johnstone, Ieke Moerdijk, Local maps of toposes Proceedings London Mathematical Society 58 (1989), 281-305.

Further refinements are in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)